One-Point Energy Correlator

• One-Point Energy Correlator (OPEC) is an observable that quantifies the angular energy distribution using a single detector insertion and process-dependent angular variables.
• It employs an energy-weighted sum over final-state radiation with a parameter (aₑ) that measures anisotropy and interpolates between quark- and hadron-dominated regimes.
• OPEC is applied across various settings—including DIS, jet substructure, and spin-dependent studies—providing a unified probe that bridges theoretical predictions with experimental observables.

The one-point energy correlator (OPEC) is an energy-flow observable that measures the angular distribution of energy with respect to a chosen reference direction through a single detector insertion. In the recent QCD literature, it appears in several closely related forms: as the expectation value of the energy flow operator in a state created by a current, as an event-shape observable in deep inelastic scattering (DIS), as an in-jet energy-flow distribution relative to a jet axis, and as a spin-dependent observable sensitive to nucleon structure. Across these settings, the common structure is an energy-weighted sum over final-state radiation, differential in an angle such as $\theta$, $\chi$, or $\tau=(1+\cos\theta)/2$, with the detailed factorization and nonperturbative content depending on the process (Riembau et al., 18 Dec 2025, Kang et al., 2 Mar 2026, Mi et al., 29 Jul 2025).

1. Foundational definitions and kinematic realizations

In field-theoretic language, the OPEC is built from the energy flow operator,

$$\mathcal{E}(\hat n)=\int_0^\infty dt \lim_{r\to\infty} r^2 n^i T_{0i}(t,r\hat n),$$

which measures energy flowing in the direction $\hat n$. For a source creating a state $|\Psi\rangle$, the one-point correlator is the expectation value $\langle \Psi|\mathcal{E}_n|\Psi\rangle$, and in collider language it corresponds to the angular distribution of energy deposited in a detector cell or calorimeter patch (Riembau et al., 18 Dec 2025).

The reference direction is process dependent. In current-induced hadron production, the OPEC measures the angular distribution of energy emitted in the final state when a vector current creates hadrons in $e^+e^-$ annihilation. In DIS at small $x$, it is defined as

$$\frac{d\Sigma_\lambda}{d\cos\theta} = \sum_h \int d\sigma^{\gamma_\lambda^*+p\to h+X}\, z_h\, \delta(\cos\theta_{hp}-\cos\theta),$$

where $\chi$0 denotes photon polarization, $\chi$1, and $\chi$2 is the angle between the target proton or nucleus and the outgoing hadron. The variable $\chi$3 is used for plotting and analysis, mapping angular information to the interval $\chi$4 (Kang et al., 2 Mar 2026).

A more abstract process-independent representation used in TMD-based analyses is

$\chi$5

where $\chi$6 is the reference axis and $\chi$7 is the angle between the detected energy and that axis. In jet substructure, the corresponding in-jet observable is written as

$\chi$8

with $\chi$9 the hadron energy, $\tau=(1+\cos\theta)/2$0 the jet energy, and $\tau=(1+\cos\theta)/2$1 the angle between the hadron and the jet axis (Fu et al., 18 Dec 2025, Mi et al., 29 Jul 2025).

These definitions all implement a one-point measurement of energy flow, but they do not define a single universal convention. The literature instead uses the same term for a family of single-detector, angle-differential, energy-weighted observables adapted to different hard processes and factorization schemes.

2. Angular structure, positivity bounds, and spinning sources

For unpolarized collisions initiated by a vector current, the OPEC takes a fully general form parameterized by a single real parameter $\tau=(1+\cos\theta)/2$2,

$\tau=(1+\cos\theta)/2$3

with $\tau=(1+\cos\theta)/2$4 the total deposited energy and $\tau=(1+\cos\theta)/2$5 the angle between the beam axis and the detector direction. The parameter $\tau=(1+\cos\theta)/2$6 quantifies the anisotropy of the angular energy distribution and obeys the strict bounds

$\tau=(1+\cos\theta)/2$7

These bounds are saturated in free theories: $\tau=(1+\cos\theta)/2$8 for free, minimally coupled fermion matter, and $\tau=(1+\cos\theta)/2$9 for free, minimally coupled scalar matter (Riembau et al., 2 Sep 2025).

In QCD, the same parameter organizes a nontrivial flow between ultraviolet and infrared regimes. At high $$\mathcal{E}(\hat n)=\int_0^\infty dt \lim_{r\to\infty} r^2 n^i T_{0i}(t,r\hat n),$$0, where quarks dominate, $$\mathcal{E}(\hat n)=\int_0^\infty dt \lim_{r\to\infty} r^2 n^i T_{0i}(t,r\hat n),$$1; at low $$\mathcal{E}(\hat n)=\int_0^\infty dt \lim_{r\to\infty} r^2 n^i T_{0i}(t,r\hat n),$$2, where pions and hadronic resonances dominate, $$\mathcal{E}(\hat n)=\int_0^\infty dt \lim_{r\to\infty} r^2 n^i T_{0i}(t,r\hat n),$$3. The paper reconstructing this flow relates the parameter directly to the longitudinal fraction of the inclusive cross section,

$$\mathcal{E}(\hat n)=\int_0^\infty dt \lim_{r\to\infty} r^2 n^i T_{0i}(t,r\hat n),$$4

and reports that existing measurements from LEP, PETRA, and SLAC support the theoretical picture; one quoted example is DELPHI at $$\mathcal{E}(\hat n)=\int_0^\infty dt \lim_{r\to\infty} r^2 n^i T_{0i}(t,r\hat n),$$5, with $$\mathcal{E}(\hat n)=\int_0^\infty dt \lim_{r\to\infty} r^2 n^i T_{0i}(t,r\hat n),$$6 (Riembau et al., 2 Sep 2025).

The spinning-source formulation extends the OPEC beyond this scalar parameterization. For vector sources, the hadronic tensor can be decomposed so that $$\mathcal{E}(\hat n)=\int_0^\infty dt \lim_{r\to\infty} r^2 n^i T_{0i}(t,r\hat n),$$7 controls the traceless tensor structure, while in helicity space the correlator becomes a convex combination of different spin projections along the detector axis. The general angular dependence is expressed in terms of Euler angles and Wigner $$\mathcal{E}(\hat n)=\int_0^\infty dt \lim_{r\to\infty} r^2 n^i T_{0i}(t,r\hat n),$$8-matrices, and the off-diagonal entries in the polarization density matrix carry a nontrivial $$\mathcal{E}(\hat n)=\int_0^\infty dt \lim_{r\to\infty} r^2 n^i T_{0i}(t,r\hat n),$$9 dependence. In this formulation, the OPEC is not a mere inclusive energy sum but a probe of angular momentum transfer and polarization structure. The same framework derives generalized sum rules and positivity bounds for arbitrary spin sources and for conserved charges beyond energy (Riembau et al., 18 Dec 2025).

3. Deep inelastic scattering at small $\hat n$0

The small-$\hat n$1 DIS formulation places the OPEC in the Color Glass Condensate (CGC) framework and treats it as an event-shape observable measuring the angular distribution of energy flow with respect to the incoming proton or nucleus. The calculation is performed in the Breit frame. The virtual photon $\hat n$2 first splits into a $\hat n$3 dipole, the dipole scatters eikonally off the gluon field of the target, and the struck quark or antiquark fragments into a hadron whose energy flow is measured relative to the incident target (Kang et al., 2 Mar 2026).

The semi-inclusive hadron production cross section is written as a convolution of fragmentation functions with partonic cross sections, while the partonic cross sections themselves depend on a hard kernel $\hat n$4 and on $\hat n$5, the Fourier transform of the dipole $\hat n$6-matrix $\hat n$7. The dipole amplitude is evolved with the rcBK equation, and for nuclei the initial saturation scale is rescaled as

$\hat n$8

with $\hat n$9 reflecting nuclear geometry uncertainties (Kang et al., 2 Mar 2026).

The central theoretical simplification is the cancellation of fragmentation functions through the momentum sum rule,

$|\Psi\rangle$0

Because the OPEC is weighted by $|\Psi\rangle$1 and summed over hadron species, all dependence on fragmentation functions exactly cancels. The resulting observable depends only on the dipole amplitude, making it a direct probe of the small-$|\Psi\rangle$2 gluonic structure and of gluon saturation (Kang et al., 2 Mar 2026).

The numerical study targets kinematics relevant to the future Electron–Ion Collider, with $|\Psi\rangle$3, $|\Psi\rangle$4--$|\Psi\rangle$5, and $|\Psi\rangle$6. The OPEC exhibits sizable nuclear suppression at small $|\Psi\rangle$7. For Gold at low $|\Psi\rangle$8 and small $|\Psi\rangle$9, the nuclear modification ratio is quoted as $\langle \Psi|\mathcal{E}_n|\Psi\rangle$0--$\langle \Psi|\mathcal{E}_n|\Psi\rangle$1; as $\langle \Psi|\mathcal{E}_n|\Psi\rangle$2 increases, $\langle \Psi|\mathcal{E}_n|\Psi\rangle$3 rises toward $\langle \Psi|\mathcal{E}_n|\Psi\rangle$4, indicating weaker suppression when saturation subsides. Suppression also becomes less pronounced at higher $\langle \Psi|\mathcal{E}_n|\Psi\rangle$5, and the transverse OPEC is typically much larger than the longitudinal contribution, so the total is dominated by transverse photon scattering. The angular variable therefore scans momentum and Bjorken-$\langle \Psi|\mathcal{E}_n|\Psi\rangle$6 scales, with small $\langle \Psi|\mathcal{E}_n|\Psi\rangle$7 corresponding to large angles and low transverse momentum, where saturation effects are strongest (Kang et al., 2 Mar 2026).

4. Back-to-back and collinear OPEC in electron–ion collisions

A complementary line of work studies the OPEC at both the back-to-back and collinear limits in electron–proton and electron–nucleus collisions. In this setting the observable is explicitly described as infrared and collinear safe because it is an energy-weighted sum over all final-state hadrons. The back-to-back limit corresponds to semi-inclusive DIS in the Breit frame with $\langle \Psi|\mathcal{E}_n|\Psi\rangle$8, or equivalently $\langle \Psi|\mathcal{E}_n|\Psi\rangle$9, while the collinear limit addresses in-jet energy correlations at small angular separation $e^+e^-$0 (Fu et al., 18 Dec 2025).

In both regimes the OPEC factorizes into hard, soft, and TMD components. In the back-to-back case it depends on TMD PDFs and on an EEC jet function, while in the collinear case it involves the EEC jet function together with global and collinear-soft functions. The EEC jet function is defined as

$e^+e^-$1

so in this formulation the observable retains explicit dependence on TMD fragmentation physics rather than canceling it (Fu et al., 18 Dec 2025).

Nuclear effects are incorporated through nuclear-modified TMD PDFs and TMD fragmentation functions. The quoted phenomenology uses nuclear PDF sets such as EPPS16 and nuclear FFs such as LIKEn 2021, with additional nonperturbative broadening modeled through $e^+e^-$2 and $e^+e^-$3 (Fu et al., 18 Dec 2025).

The resulting nuclear modification factor,

$e^+e^-$4

is reported to be significantly below unity in the small-angle or low-$e^+e^-$5 region. In the back-to-back case, $e^+e^-$6 is approximately $e^+e^-$7 at small $e^+e^-$8 and rises with increasing $e^+e^-$9. In the collinear in-jet case, the OPEC peaks at small $x$0, and the nuclear modification factor is suppressed to $x$1--$x$2 in the small-$x$3 region, with suppression weakening at larger $x$4. The back-to-back observable is sensitive to the convolution of initial-state TMD PDFs and final-state TMD FFs, whereas the collinear angular distribution is dominated by final-state nuclear effects and can be interpreted as a probe of jet evolution and hadronization timescales (Fu et al., 18 Dec 2025).

5. One-point energy correlator inside jets

The in-jet OPEC was introduced as a new jet observable designed to characterize the energy deposition at a specific angle relative to the jet axis. Its small-angle limit is directly connected to transverse momentum dependent physics through

$x$5

so measuring the angular distribution of energy flow inside a jet is equivalent, in the collinear regime, to probing intrajet transverse momentum distributions (Mi et al., 29 Jul 2025).

The factorization is derived in Soft-Collinear Effective Theory using semi-inclusive TMD fragmenting jet functions. The EC jet function is written as a convolution over transverse momentum and hadron momentum fraction, and the semi-inclusive TMD fragmenting jet function further factorizes into a hard matching coefficient $x$6, a TMD fragmentation function $x$7, and an in-jet soft function $x$8. In $x$9 collisions this formulation depends on TMD fragmentation but not on TMD parton distributions, which is why the observable is proposed as a new way to access gluon TMD fragmentation functions (Mi et al., 29 Jul 2025).

Large logarithms are resummed at high perturbative accuracy. The analysis reports resummation up to NNLL accuracy for global logarithms and LL accuracy for non-global logarithms, with nonperturbative corrections implemented through the $$\frac{d\Sigma_\lambda}{d\cos\theta} = \sum_h \int d\sigma^{\gamma_\lambda^*+p\to h+X}\, z_h\, \delta(\cos\theta_{hp}-\cos\theta),$$0 prescription,

$$\frac{d\Sigma_\lambda}{d\cos\theta} = \sum_h \int d\sigma^{\gamma_\lambda^*+p\to h+X}\, z_h\, \delta(\cos\theta_{hp}-\cos\theta),$$1

The normalized EC jet function is found to exhibit significantly reduced dependence on the factorization scale and to be primarily sensitive to the jet scale. After inclusion of nonperturbative effects, the numerical predictions are compared with PYTHIA 8, and the comparison is described as excellent at small $$\frac{d\Sigma_\lambda}{d\cos\theta} = \sum_h \int d\sigma^{\gamma_\lambda^*+p\to h+X}\, z_h\, \delta(\cos\theta_{hp}-\cos\theta),$$2 in both $$\frac{d\Sigma_\lambda}{d\cos\theta} = \sum_h \int d\sigma^{\gamma_\lambda^*+p\to h+X}\, z_h\, \delta(\cos\theta_{hp}-\cos\theta),$$3 and $$\frac{d\Sigma_\lambda}{d\cos\theta} = \sum_h \int d\sigma^{\gamma_\lambda^*+p\to h+X}\, z_h\, \delta(\cos\theta_{hp}-\cos\theta),$$4 collisions. The same study emphasizes that quark TMDFFs dominate the $$\frac{d\Sigma_\lambda}{d\cos\theta} = \sum_h \int d\sigma^{\gamma_\lambda^*+p\to h+X}\, z_h\, \delta(\cos\theta_{hp}-\cos\theta),$$5 case, while the $$\frac{d\Sigma_\lambda}{d\cos\theta} = \sum_h \int d\sigma^{\gamma_\lambda^*+p\to h+X}\, z_h\, \delta(\cos\theta_{hp}-\cos\theta),$$6 case is more sensitive to gluon TMDFFs (Mi et al., 29 Jul 2025).

Taken together with the small-$$\frac{d\Sigma_\lambda}{d\cos\theta} = \sum_h \int d\sigma^{\gamma_\lambda^*+p\to h+X}\, z_h\, \delta(\cos\theta_{hp}-\cos\theta),$$7 DIS construction, this literature shows that fragmentation independence is not a universal feature of the OPEC. In DIS at small $$\frac{d\Sigma_\lambda}{d\cos\theta} = \sum_h \int d\sigma^{\gamma_\lambda^*+p\to h+X}\, z_h\, \delta(\cos\theta_{hp}-\cos\theta),$$8, the momentum sum rule removes fragmentation functions exactly; in TMD and in-jet formulations, the observable instead keeps explicit dependence on TMD fragmentation functions or on the EEC jet function.

6. Spin-dependent OPEC and nucleon transversity

A separate development uses the OPEC to access the nucleon transversity distribution $$\frac{d\Sigma_\lambda}{d\cos\theta} = \sum_h \int d\sigma^{\gamma_\lambda^*+p\to h+X}\, z_h\, \delta(\cos\theta_{hp}-\cos\theta),$$9 in transversely polarized proton–proton collisions. In this setting the observable measures the angular distribution of energy flow inside a jet, differential in polar and azimuthal angles relative to the jet axis, and is written as

$\chi$00

The corresponding single-spin asymmetry is

$\chi$01

The characteristic modulation is a clean $\chi$02 angular dependence, with $\chi$03 the azimuthal angle of the transverse spin vector and $\chi$04 the azimuthal angle of the energy-flow measurement inside the jet (Gao et al., 19 Sep 2025).

The factorized description involves ordinary PDFs, the chiral-odd transversity PDF $\chi$05, hard coefficients, and energy-weighted fragmenting jet functions. The spin-dependent structure contains the transversity distribution multiplied by a Collins-type fragmentation contribution and a hard factor, so the OPEC realizes a Collins mechanism in an energy-weighted, angle-differential form (Gao et al., 19 Sep 2025).

This approach is presented as complementary to traditional hadron-in-jet measurements based on the leading hadron transverse momentum $\chi$06. Because the OPEC integrates over all hadrons and all $\chi$07, it probes the asymmetry over a much wider kinematic range in the angular scale $\chi$08 than traditional $\chi$09 measurements and accesses smaller angular scales. The observable is also described as infrared and collinear safe and less vulnerable to hadronization uncertainties than exclusive leading-hadron observables. The proposed applications include RHIC and the future Electron–Ion Collider, where the method is intended to enable systematic cross-checks and universality tests of spin-dependent fragmentation (Gao et al., 19 Sep 2025).

7. Relation to higher-point correlators and generalized energy-correlation frameworks

The OPEC is the simplest member of the broader family of energy correlators. In the standard hierarchy, the one-point function measures the mean energy flow in a single direction, the two-point correlator is the energy–energy correlator, and three-point or higher-point correlators resolve multiprong geometry. This distinction is operationally important: the OPEC captures a gross angular energy profile, whereas higher-point correlators are sensitive to pairwise or multiparticle structure, such as the three-prong decay kinematics exploited in boosted top-quark studies (Holguin et al., 2022).

An alternative use of the term appears in the literature on analytically continued projected $\chi$10-point energy correlators. There, the projected $\chi$11-point correlator is analytically continued to non-integer $\chi$12, and the limit $\chi$13 is referred to as the OPEC. In that limit the observable relates to hadron multiplicity and probes the small-momentum-fraction behavior of splitting functions. The cumulative distribution obeys an approximate scaling law,

$\chi$14

with DGLAP governing the behavior for $\chi$15, while the small-$\chi$16 limit requires BFKL resummation. The same work reports a computational reduction from naive $\chi$17 scaling to $\chi$18 through recursion and then to a FastEEC implementation based on subjets; limiting the number of subjets to $\chi$19 is said to be sufficient for percent-level precision. Applied to CMS Open Data, the extracted scaling exponents agree with DGLAP away from $\chi$20 and saturate to a value matching the BFKL anomalous dimension as $\chi$21 (Budhraja et al., 2024).

A neighboring development is the framework of Energy Weighted Observable Correlations (EWOCs), which generalizes pairwise energy correlators beyond angles by using subjets and arbitrary pairwise observables such as mass or formation time. That framework explicitly focuses on two-point and higher-point correlations and states that there is no direct mention or generalization of a one-point OPEC observable. This suggests that the OPEC occupies a limiting but structurally important position within the broader program of energy-weighted correlation measurements: it is the minimal energy-flow observable, while much of the recent generalization effort has proceeded through pairwise and multipoint constructions (Alipour-fard et al., 28 Jan 2025).